J. Phys. A: Math. Gen. 31 (1998) 3297–3300. Printed in the UK PII: S0305-4470(98)89896-X
On exact Foldy–Wouthuysen transformation
A G Nikitin
Institute of Mathematics, National Academy of Sciences of the Ukraine, Tereshchenkivs’ka Street 3, Kiev-4, Ukraine
Received 9 December 1997
Abstract. New classes of external fields are found for which the exact Foldy–Wouthuysen transformation for the Dirac equation exists.
The Foldy–Wouthuysen transformation (FWT) [1] is one of the corner stones of relativistic quantum mechanics. It presents a straightforward and convenient way to obtain an adequate physical interpretation for relativistic wave equations. The very existence of an exact FWT for a considered relativistic problem justifies its quantum mechanical treatment, inasmuch as in this case transitions between positive and negative energy states are forbidden. Following [2] we say that the related Dirac sea is stable, and so it is possible to restrict ourselves to positive energy solutions.
The known problems, which admit the exact FWT, include minimal [3] and anomalous [4] interactions of spin-12 particles with a time-independent magnetic field, anomalous interaction of such particles with a time-independent electric field and with a pseudo-scalar field [2]. A specific generalization of the FWT for the Dirac equation with a half sum of scalar and zero component of four-vector potentials was proposed in [5]. Moreover, the analysis present in papers [2, 5] is based on custodial supersymmetry of the Dirac equation.
The aim of this paper is to present new classes of problems admitting the exact FWT.
Let us start with the Dirac Hamiltonian for a free particle
H=cγ0γapa+γ0mc2 (1)
whereγ0,γa (a=1,2,3) are the Dirac matrices with diagonalγ0. The FWT reduces (1) to the diagonal form
H→H0=U H U†=γ0√
H2 (2)
hereH2=p2c2+m2c4, and the related operatorU is found in paper [1]. In our notations it has the form
U=exp γapa
p arctanp mc
.
It was proposed by Eriksen and Kolsrud [4] to use the two-step transformation (2) with U=U2U1 U1= 1
√2(1+I ) U2= 1
√2(1+γ0I ) (3)
0305-4470/98/143297+04$19.50 c 1998 IOP Publishing Ltd 3297
3298 A G Nikitin
where = H /(H2)1/2 is the sign energy operator and I = iγ5γ0 is a unitary involution anticommuting withH andγ0
I†I =I I†=I2=1 I H = −H I I γ0= −γ0I. (4) It is the operatorU given in (3) which, in contrast with the original Foldy–Wouthuysen operator, admits the direct generalization for the case of anomalous interaction with a time- independent magnetic field [4].
We notice that relations (4) can be used to search for new types of external fields such that the related Dirac Hamiltonian for an interacting particle admits the exact FWT. The main idea which enables us to achieve this goal is that the corresponding involutionI can be constructed using discrete symmetries of the Dirac equation studied in [6].
To demonstrate new possibilities which arise in this way let us consider the Dirac Hamiltonian for a particle interacting with an electrostatic field
H=cγ0γapa+γ0mc2+e
cA0. (5)
It is generally accepted that this Hamiltonian admits only an approximate FWT [1].
Here we shall show that operator (5) also admits the exact FWT provided it has no zero eigenvalues and A0 is an odd function of x: A0(−x) = −A0(x). Familiar examples of oddA0 are the potentials of the constant and dipole electric fields, when
A0=qex·E and A0= qe
|x−a|− qe
|x+a| respectively (E andaare constant vectors).
The corresponding involutionI is expressed via the space reflection operatorP I =iγ5γ0P PΨ(x0,x)=γ0Ψ(x0,−x). (6) Substituting (5) and (6) into (3) we obtain
U H U†=H0=γ0
p2c2+m2c4+e2
c2A20+2mceγ0A0+P{eA0,σ·p} 1/2
(7) whereσ=iγ×γ/2.
Formula (7) presents probably the first known example of the exact FWT for the Dirac Hamiltonian including a minimal interaction with an external electric field. ExpandingH0in a power series of 1/cand neglecting the terms of orders 1/c3, we come to the approximate quasirelativistic Hamiltonian
HQR=γ0
m2c4+ p2 2m− p4
8m3c2 +eP{σ·p, A0} 2mc2
−e{p2, A0} 4m2c +e
cA0
which is unitary equivalent to the corresponding Hamiltonian found by Foldy and Wouthuysen
W HQRW†=γ0
mc2+ p2 2m− p4
8m3c2
+eA0− e
8m2c2[σ·(π×E−E×π)+divE]
where
W =exp
−γ0Pσ·p 2mc
and Ea= −∂A0
∂xa
.
Finally, let us consider the Hamiltonian of spin-12 particle of the following general form H=cγ0γaπa+γ0(mc2+V )+γ5ϕ+ k
m(γ5γaHa+iγaEa)+eA0. (8)
On exact Foldy–Wouthuysen transformation 3299 Here πa = pa −eAa(x), A0 and Aa are components of a vector-potential, Ha and Ea
are components of the vectors of magnetic and electric fields, V and ϕ are scalar and pseudoscalar potentials.
Hamiltonian (8) includes minimal and anomalous interactions with an electromagnetic field, and also scalar and pseudoscalar interactions. To construct the exact FWT for this Hamiltonian, let us search for involutionsI (satisfying (4)) in the form
I =0ARA (9)
(no sum over A) where 0A are numeric matrices, the multi-index A takes the values A = 1,2,3,12,32,23,123, RA are operators of the discrete transformations of spatial variablesRAψ(x0,x)=ψ(x0, rAx), and
r1x=(−x1, x2, x3) r2x=(x1,−x2, x3) r3x=(x1, x2,−x3) r12x= −r3x r23x= −r1x r31x= −r2x r123x= −x.
Requiring the anticommutativity of operators (7) with the free particle Hamiltonian (1), it is not difficult to specify explicit forms of matrices0A and find all possible involutions (9) in the form
I =γ0γaRa a =1,2,3 (10a)
I =iγaRbc a6=b, a6=c, b6=c (10b)
I =iγ5R123. (10c)
Substituting (8) and (10a)–(10c) into (2) we come to the following conditions (11a)–(11c)forA0,A=(A1, A2, A3) V andϕ, respectively
A0(rax)= −A0(x) A(rax)=raA(x) V (rax)=V (x)
ϕ(rax)=ϕ(x) (11a)
A0(rabx)= −A0(x) A(rabx)=rabA(x) V (rabx)=V (x)
ϕ(rabx)= −ϕ(x) (11b)
A0(−x)= −A0(x) A(−x)= −A(x) V (−x)=V (x)
ϕ(−x)=ϕ(x). (11c)
Thus, we have found the exact FWT (generated by operators (3) and (10)) for the generalized Dirac Hamiltonian (8) which includes minimal, anomalous, scalar and pseudoscalar interactions with external fields. A sufficient condition for the existence of this transformation is that the related potentials satisfy at least one of the relations (11a), (11b), or(11c), and that the corresponding Hamiltonian has no zero eigenvalues.
Relations (11) define a wide class of potentials, including such interesting ones as the constant electric and magnetic fields, electric dipole field, electric field generated by two infinite charged strings with opposite charges, magnetic field generated by an infinite straight conductor and many, many others. Considering the superpositions of these fields, it is possible to generate examples of realistic physical systems with different types of symmetries enumerated in (11).
We see that using discrete involutive symmetries of the Dirac equation, it is possible to extend sufficiently the class of external fields admitting the exact FWT. Algebraic structures of these symmetries were studied in papers [6].
An exact FWT for a two-body equation is found in [7]. For a survey of FWT for higher spin and two particle equations refer to [8].
3300 A G Nikitin Acknowledgment
This work was supported by the Ukrainian DFFD foundation through grant No 1.4/356.
References
[1] Foldy L L and Wouthuysen S 1950 Phys. Rev. 58 29
[2] Moreno M, Martinez R and Zentella A 1990 Mod. Phys. Lett. 5 949 [3] Case K M 1954 Phys. Rev. 95 1323
[4] Eriksen E and Kolsrud M 1960 Nuovo Cimento 18 1
[5] Moreno M and Mendes-Moreno R M 1992 Proc. Workshops on High Energy Phenomenology ed M A Perez and R Huerta (Singapore: World Scientific) p 365
[6] Niederle J and Nikitin A G 1997 J. Phys. A: Math. Gen. 30 999 Niederle J and Nikitin A G 1997 J. Nonlin. Math. Phys. 4 436 [7] Nikitin A G and Tretynyk V V 1997 Int. J. Mod. Phys. 12 4369
[8] Fushchich W I and Nikitin A G 1994 Symmetries of Equations of Quantum Mechanics (New York: Allerton)
